As far as I can tell, your phantom cannot benefit from your racial traits. However as a player this is throwing me for a loop, because your phantom is an aspect of your character, even moreso with the Fractured Mind Spiritualist archtype.
"Most spiritualists harbor the spirits of the deceased in their psyches, but a small number of them—known as fractured minds—draw their powers instead from a fraction of their own souls that resonates with extremely powerful emotions. These spiritualists’ phantoms are not spiritual allies, but rather extensions of the fractured minds’ own inner thoughts and emotions."
So, as a Kitsune, I took the alternate racial trait, Gregarious: "Whenever you successfully use Diplomacy to win over an individual, that creature takes a –2 penalty to resist any of your Charisma-based skill checks for the next 24 hours." Default when I'm doing my Lust phantom's actions I'm thinking about the debuff, even though I'm pretty sure it doesn't apply.
Maybe it's simply because most of my GMs generally let me control any minions/companions/cohorts I happen to have, trusting me to roleplay them appropriately (animals are generally too dumb to move tactically according to the book*, generally taking the fastest route to a spot they are commanded to go to for example). But I'm struggling a bit managing all the applications to my Spiritualist and Phantom because they do a lot of the same face character things.
*This is verbatim from the Playtest book, however I do know that within 1e books it is generally labeled as the GM's responsibility to control most non-PCs, including companions/cohorts. Animate Dead makes specific mention to needing to verbally command your minions which was one of my last dealings with these rules prior to now, so I may be wrong.
I'm not sure what your question is, but you are correct that your phantom won't get your racial bonus. Even if it comes from 'a fraction of your soul' it isn't you. It is it's own creature with it's own bonuses and abilities.
If it helps, just assume that the fraction of your soul that it comes from doesn't include the gregarious part.
Dave got it.
Having never really played a companion character before a lot of this is new. The other bit is whether my phantom can still do stuff like make skill checks, or in theory possess me whioe in my consciousness. Again, I kind of assume the answer to be no. Must be manifested to interact with the world, even socially.
The spiritualist class tells you what abilities your phantom has. Have you fully read the class?
A fully manifested phantom is a creature and can take actions and do things according to it's abilities.
A phantom can be harbored in your consciousness, which gives you the benefits from the shared consciousness class feature, but that is all you get.
It can be partially manifested, which give you the benefits from the bonded manifestation ability.
If your phantom is on the ethereal plane it pretty much doesn't exist for that time as far as you are concerned, other than your abilities to bring it back.